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Laurent Lafleche

Publications and source records attributed to Laurent Lafleche.

At least 19 recordsLinked to original sources
arXiv2026

Commutator Estimates for Low-Temperature Fermi Gases

We investigate the semiclassical regularity of thermal equilibria in the presence of a harmonic potential at low temperature; that is, we obtain the asymptotic behavior of the Schatten norms of commutators of the one-body operators associated with these equilibria and the position and momentum operators. We also obtain upper bounds in the magnetic field case for the Fock-Darwin Hamiltonian. Our estimates, in particular, allow us to observe several regimes depending on the joint behavior of the Planck constant, the temperature, and the strength of the magnetic field.

math-ph↗
arXiv2025

Uncertainty Relation for the Wigner-Yanase Skew Information and Quantum Sobolev Inequalities

This note explores uncertainty inequalities for quantum analogues of the Fisher information including the Wigner-Yanase skew information, and their connection to the quantum Sobolev inequalities proved by the author in [Journal of Functional Analysis, 286 (10) 2024]. Some additional inequalities concerning commutators are derived and others are left as open problems.

math-ph↗
arXiv2025

Commutator Estimates and Quantitative Local Weyl's Law for Schrödinger Operators with Non-Smooth Potentials

We analyze semi-classical Schrödinger operators with potentials of class $C^{1,1/2}$ and establish commutator estimates for the associated projection operators in Schatten norms. These are then applied to prove quantitative versions of the local and phase space Weyl laws in $L^p$ spaces. We study both non-interacting, and interacting particle systems. In particular, we are able to treat the case of the minimizers of the Hartree energy in the case of repulsive singular pair interactions such as the Coulomb potential.

math-ph↗
arXiv2024

Semiclassical Limit of the Bogoliubov-de Gennes Equation

In this paper, we rewrite the time-dependent Bogoliubov$\unicode{x2013}$de Gennes equation in an appropriate semiclassical form and establish its semiclassical limit to a two-particle kinetic transport equation with an effective mean-field background potential satisfying the one-particle Vlasov equation. Moreover, for some semiclassical regimes, we obtain a higher-order correction to the two-particle kinetic transport equation, capturing a nontrivial two-body interaction effect. The convergence is proven for $C^2$ interaction potentials in terms of a semiclassical optimal transport pseudo-metric. Furthermore, combining our current results with the results of Marcantoni et al. [arXiv:2310.15280], we establish a joint semiclassical and mean-field approximation of the dynamics of a system of spin-$\frac{1}{2}$ Fermions by the Vlasov equation in some negative order Sobolev topology.

math-ph↗
arXiv2024

Enhanced Stability in Quantum Optimal Transport Pseudometrics: From Hartree to Vlasov-Poisson

In this paper we establish almost-optimal stability estimates in quantum optimal transport pseudometrics for the semiclassical limit of the Hartree dynamics to the Vlasov-Poisson equation, in the regime where the solutions have bounded densities. We combine Golse and Paul's method from [Arch. Ration. Mech. Anal. 223:57-94, 2017], which uses a semiclassical version of the optimal transport distance and which was adapted to the case of the Coulomb and gravitational interactions by the second author in [J. Stat. Phys. 177:20-60, 2019], with a new approach developed by the first author in [Arch. Ration. Mech. Anal. 244:27-50, 2022] to quantitatively improve stability estimates in kinetic theory.

math.AP↗
arXiv2023

Quantum Optimal Transport and Weak Topologies

Several extensions of the classical optimal transport distances to the quantum setting have been proposed. In this paper, we investigate the pseudometrics introduced by Golse, Mouhot and Paul in [Commun Math Phys 343:165-205, 2016] and by Golse and Paul in [Arch Ration Mech Anal 223:57-94, 2017]. These pseudometrics serve as a quantum analogue of the Monge-Kantorovich-Wasserstein distances of order $2$ on the phase space. We prove that they are comparable to negative Sobolev norms up to a small term due to a positive "self-distance" in the semiclassical approximation, which can be bounded above using the Wigner-Yanase skew information. This enables us to improve the known results in the context of the mean-field and semiclassical limits by requiring less regularity on the initial data.

math.AP↗
arXiv2023

On Quantum Sobolev Inequalities

We investigate the quantum analogue of the classical Sobolev inequalities in the phase space, with the quantum Sobolev norms defined in terms of Schatten norms of commutators. These inequalities provide an uncertainty principle for the Wigner-Yanase skew information, and also lead to new bounds on the Schatten norms of the Weyl quantization in terms of its symbol. As an intermediate tool, we obtain the analogue of Hardy-Littlewood-Sobolev's inequalities for a semiclassical analogue of the convolution, and introduce quantum Besov spaces. Explicit estimates are obtained on the optimal constants.

math-ph↗
arXiv2023

Optimal Semiclassical Regularity of Projection Operators and Strong Weyl Law

Projection operators arise naturally as one-particle density operators associated to Slater determinants in fields such as quantum mechanics and the study of determinantal processes. In the context of the semiclassical approximation of quantum mechanics, projection operators can be seen as the analogue of characteristic functions of subsets of the phase space, which are discontinuous functions. We prove that projection operators indeed converge to characteristic functions of the phase space and that in terms of quantum Sobolev spaces, they exhibit the same maximal regularity as characteristic functions. This can be interpreted as a semiclassical asymptotic on the size of commutators in Schatten norms. Our study answers a question raised in [J. Chong, L. Lafleche, C. Saffirio, arXiv:2103.10946 [math.AP]] about the possibility of having projection operators as initial data. It also gives a strong convergence result in Sobolev spaces for the Weyl law in phase space.

math-ph↗
arXiv2023

From many-body quantum dynamics to the Hartree-Fock and Vlasov equations with singular potentials

We obtain the combined mean-field and semiclassical limit from the $N$-body Schrödinger equation for fermions interacting via singular potentials. To obtain the result, we first prove the uniformity in Planck's constant $h$ propagation of regularity for solutions to the Hartree$\unicode{x2013}$Fock equation with singular pair interaction potentials of the form $\pm |x-y|^{-a}$, including the Coulomb and gravitational interactions. In the context of mixed states, we use these regularity properties to obtain quantitative estimates on the distance between solutions to the Schrödinger equation and solutions to the Hartree$\unicode{x2013}$Fock and Vlasov equations in Schatten norms. For $a\in(0,1/2)$, we obtain local-in-time results when $N^{-1/2} \ll h \leq N^{-1/3}$. In particular, it leads to the derivation of the Vlasov equation with singular potentials. For $a\in[1/2,1]$, our results hold only on a small time scale, or with an $N$-dependent cutoff.

math.AP↗
arXiv2022

Global-in-time Semiclassical Regularity for the Hartree-Fock Equation

For arbitrarily large times $T>0$, we prove the uniform-in-$\hbar$ propagation of semiclassical regularity for the solutions to the Hartree$\unicode{x2013}$Fock equation with singular interactions of the form $V(x)=\pm\,|x|^{-a}$ where $a\in(0,\frac12)$. As a byproduct of this result, we extend to arbitrarily long times the derivation of the Hartree$\unicode{x2013}$Fock and the Vlasov equations from the many-body dynamics provided in [J. Chong, L. Lafleche, C. Saffirio: arXiv:2103.10946 (2021)].

math.AP↗
arXiv2022

On the semiclassical regularity of thermal equilibria

We study the regularity properties of fermionic equilibrium states at finite positive temperature and show that they satisfy certain semiclassical bounds. As a corollary, we identify explicitly a class of positive temperature states satisfying the regularity assumptions of [J.J. Chong, L. Lafleche, C. Saffirio: arXiv:2103.10946 (2021)].

math-ph↗
arXiv2022

On the $L^2$ Rate of Convergence in the Limit from the Hartree to the Vlasov$\unicode{x2013}$Poisson Equation

Using a new stability estimate for the difference of the square roots of two solutions of the Vlasov$\unicode{x2013}$Poisson equation, we obtain the convergence in the $L^2$ norm of the Wigner transform of a solution of the Hartree equation with Coulomb potential to a solution of the Vlasov$\unicode{x2013}$Poisson equation, with a rate of convergence proportional to $\hbar$. This improves the $\hbar^{3/4-\varepsilon}$ rate of convergence in $L^2$ obtained in [L.~Lafleche, C.~Saffirio: Analysis & PDE, to appear]. Another reason of interest of this paper is the new method, reminiscent of the ones used to prove the mean-field limit from the many-body Schrödinger equation towards the Hartree$\unicode{x2013}$Fock equation for mixed states.

math.AP↗
arXiv2021

Fractional Hypocoercivity

This paper is devoted to kinetic equations without confinement. We investigate the large time behaviour induced by collision operators with fat tailed local equilibria. Such operators have an anomalous diffusion limit. In the appropriate scaling, the macroscopic equation involves a fractional diffusion operator so that the optimal decay rate is determined by a fractional Nash type inequality. At kinetic level we develop an $\mathrm L^2$-hypocoercivity approach and establish a rate of decay compatible with the fractional diffusion limit.

math.AP↗
arXiv2021

Strong semiclassical limit from Hartree and Hartree-Fock to Vlasov-Poisson equation

We consider the semiclassical limit from the Hartree to the Vlasov equation with general singular interaction potential including the Coulomb and gravitational interactions, and we prove explicit bounds in the strong topologies of Schatten norms. Moreover, in the case of fermions, we provide estimates on the size of the exchange term in the Hartree-Fock equation and also obtain a rate of convergence for the semiclassical limit from the Hartree-Fock to the Vlasov equation in Schatten norms. Our results hold for general initial data in some Sobolev space and any fixed time interval.

math-ph↗
arXiv2020

Global Semiclassical Limit from Hartree to Vlasov Equation for Concentrated Initial Data

We prove a quantitative and global in time semiclassical limit from the Hartree to the Vlasov equation in the case of a singular interaction potential in dimension $d\geq 3$, including the case of a Coulomb singularity in dimension $d=3$. This result holds for initial data concentrated enough in the sense that some space moments are initially sufficiently small. As an intermediate result, we also obtain quantitative semiclassical bounds on the space and velocity moments of even order and the asymptotic behavior of the spatial density due to dispersion effects.

math.AP↗
arXiv2020

Instability for Axisymmetric Blow-up Solutions to Incompressible Euler Equations

It is still not known whether a solution to the incompressible Euler equation, endowed with a smooth initial value, can blow-up in finite time. In [{\em Comm. Math. Phys.}, 378:557--568, 2020] it has been shown that, if it exists, such a solution becomes linearly unstable close to the blow-up time. In this paper, we show that the same phenomenon holds even in the more rigid axisymmetric case. To obtain this result, we first prove a blow-up criterion involving only the toroidal component of the vorticity. The instability of blow-up profiles is also investigated.

math.AP↗
arXiv2019

Hypocoercivity and sub-exponential local equilibria

Hypocoercivity methods are applied to linear kinetic equations without any space confinement, when local equilibria have a sub-exponential decay. By Nash type estimates, global rates of decay are obtained, which reflect the behavior of the heat equation obtained in the diffusion limit. The method applies to Fokker-Planck and scattering collision operators. The main tools are a weighted Poincaré inequality (in the Fokker-Planck case) and norms with various weights. The advantage of weighted Poincaré inequalities compared to the more classical weak Poincaré inequalities is that the description of the convergence rates to the local equilibrium does not require extra regularity assumptions to cover the transition from super-exponential and exponential local equilibria to sub-exponential local equilibria.

math.AP↗